166 HlSTOEY OF THE TfiEOKY OF NtJMBEBS. [CHAP. VI W. Stammer41 noted that n/p^Q.^ . . . dx implies 2(10*-!)=^... a*. P J. B. Sturm42 used this result to explain the conversion of decimal into ordinary fractions without the use of series. M. Collins43 stated that, if we multiply any decimal fraction having m digits in its period by one with n digits, we obtain a product with Qmn digits in its period if m is prime to n, but with n(10m— 1) digits if n is divisible by m. J. E. Oliver44 proved the last theorem. If x'/x gives a periodic fraction to the base a with a period of £ figures, then a*=l (mod x) and conversely. The product of the periodic fractions for x'/x, . . . , z'/z with period lengths £,..., f has the period length where M(x, . . . , 2) is the 1. c. m. of x, . . . , z. He examined the cases in which the first factor hi the formula is expressible in terms of £,..., f. Fr. Heime46 and M. Pokorny46 gave expositions without novelty. Suffield47 gave the more important rules for periodic decimals and indicated the close connection with the method of synthetic division. W. H. H. Hudson48 called d a proper prime if the period for n/d has d— I digits. If the period for r/p has n = (p — 1)/X digits, there are X periods for p. The sum of the digits in the period for a proper prime p is 9(p — 1)/2. If 1/p has a period of 2n digits, the sum of corresponding digits in the two half periods is 9, and this holds also if p is composite but has no factor dividing 10n— 1 [Midy27]. If lOp-f 1 is a proper prime, each digit 0, 1, . . . , 9 occurs p tunes in its period. If a, b are distinct primes with periods of a, /3 digits, the number of digits in the period for ab is the 1. c. m. of a, ft [Bernoulli8]. Let p have a period of n digits and l/p = k/(10n — l). Let m be the least integer for which is an integer; then l/p* has a period of mn digits. "Archiv Math. Phys., 27, 1856, 124. "Ibid., 33, 1859, 94-95. "Math. Monthly (ed., Runkle), Cambridge, Mass., 1, 1859, 295. "Ibid., 345-9. 46Ueber relative Prim- und correspondirende Zahlen, primitive und sekundare Wurzeln und periodische Decimalbriiche, Progr., Berlin, 1860, 18 pp. 4flUeber einige Eigenschaften periodischer Dezimalbriiche, Prag, 1864. 47Synthetic division in arithmetic, with some introductory remarks on the period of circulating decimals, 1863, pp. iv-f-19. "Oxford, Cambridge and Dublin Messenger of Math., 2, 1864, 1-6. Glaisher78 atrributed this useful anonymous paper to Hudson.